Asymptotic analysis of the spectrum of a quantum waveguide with a wide Neumann “window” in the light of mechanics of cracks

Various asymptotic expansions are derived for eigenvalues in the discrete spectrum of the boundary-value problem for the Laplace operator in the unit strip with the Dirichlet condition on its lateral sides everywhere with exception of an interval with length $2\ell>0$ where the Neumann condition is imposed (a planar quantum waveguide with the “window”). Since the total multiplicity of the discrete spectrum grows indefinitely as $\ell\rightarrow+\infty$, there exists a sequence of the critical lengths $\{\ell^\ast_m\}$, for which the problem operator enjoys the threshold resonance. This phenomenon is characterized by the existence of a nontrivial bounded solution, that is, either trapped, or almost standing wave, and provides miscellaneous near-threshold spectral anomalies. The quality of the threshold resonances is examined and asymptotic formulas for the values $\ell^\ast_m$ are obtained for large numbers $m$. The analysis is systematically performed by means of methods from fracture mechanics.